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Reconstructing the Orientation Distribution of Actin
Filaments in the Lamellipodium of Migrating
Keratocytes from Electron Microscopy
Tomography Data
Julian Weichsel,1,2 Edit Urban,3 J. Victor Small,3 Ulrich S. Schwarz1,2*
� AbstractMigration of motile cells on flat substrates is usually driven by the polymerizationof a flat actin filament network. Theoretical models have made different predic-tions regarding the distribution of the filament orientation in the lamellipodiumwith respect to the direction of motion. Here we show how one can automaticallyreconstruct the orientation distribution of actin filaments in the lamellipodium ofmigrating keratocytes from electron microscopy tomography data. We use two dif-ferent image analysis methods, an algorithm which explicitly extracts an abstractnetwork representation and an analysis of the gray scale information based on thestructure tensor. We show that the two approaches give similar results, both forsimulated data and for electron microscopy tomography data from migrating kera-tocytes. For the lamellipodium at the leading edge of fast moving cells, we find anorientation distribution that is peaked at 135/235 degrees. For the lamellipodiumat the leading edge of slow moving cells as well as for the lamellipodium atthe flanks of fast moving cells, one broad peak around 0 degree dominates thedistribution. ' 2012 International Society for Advancement of Cytometry
Because most optical techniques up to date are not able
to resolve the required details, electron microscopy (EM) is
the most important type of experimental data which helps to
understand the spatial organization of the lamellipodium
[12]. Pioneering work by Borisy and coworkers suggested that
the lamellipodium is organized in a surprisingly regular man-
ner, with frequent branching junctions leading to an overall
criss-cross pattern [13,14]. Recently, it has been shown with
advanced EM techniques that the organization of the lamelli-
podium might be more variable, with less branching and
more three-dimensional crosslinking [10,11,15]. The newly
available EM data now open up the perspective to achieve
more reliable and quantitative information than formerly pos-
sible, and to make interesting comparison with theoretical pre-
dictions.
In the past, different theoretical models have made inter-
esting predictions for the spatial structure of the lamellipo-
dium. Modeling the polymerization of a single filament
against an external force yielded an optimum filament orienta-
tion for fastest growth at around 488 [16]. Later, Maly and
Borisy suggested that two competing patterns exist due to a
competition of branching and orientation dependent capping
of filaments, with orientations peaked at 135/235 and 170/
0/270 degrees [17]. Motivated by the available EM evidence,
it was concluded that a pattern with peaks at 135/235
degrees dominates in lamellipodia under physiological condi-
tions, as also suggested by stochastic network simulations
[18]. Introducing an orientation dependent branching rate in
the model yielded similar 135/235 degrees patterns as before,
but with subdominant patterns adding more features to the
orientation distribution histograms [19]. Recently it has been
shown that the competition between the 170/0/270 and
135/235 degrees patterns leads to bistability and hysteresis
effects [20]. Thus, these theoretical findings are able to explain
puzzling experimental results including hysteresis effects in
actin networks grown in vitro against an atomic force micro-
scope cantilever [21].
To evaluate the experimental data quantitatively, appro-
priate image processing methods have to be developed. So far
different routines have been used to deduce information on
the network architecture of biological samples, like for
instance astrocyte cell clusters [22] and intermediate filament
networks [23]. Also actin filament networks have been ana-
lyzed and classified according to their structural organization
[24,25]. Some approaches focused specifically on the actin
network orientation from microscopy images of lamellipodia.
A combination of filament edge segmentation and subsequent
Radon transform [17,26] as well as a network model based
image analysis [27] led to the conclusion that the dominant
orientation distribution peaked at around 135/235 degrees is
the only relevant pattern in the physiological context. How-
ever, recently a correlated measurement of the leading edge ve-
locity and subsequent manual orientation analysis of ran-
domly selected single filaments demonstrated a substantial
increase in the relative number of filaments oriented more
parallel to the leading edge after transitions from protrusion
to pause [28].
Here, we refine and automatize this correlated measure-
ment approach by analyzing several two-dimensional (2D)
slices of three-dimensional (3D) tomographic EM data of the
lamellipodium network. To evaluate the orientation distribu-
tion of actin filament networks on EM images of the lamelli-
podium of motile keratocyte cells, we will rely on two different
data analysis strategies. First, we will extract an abstract 2D
actin network representation, consisting of nodes connected
by straight line segments. This algorithm was customized for
our specific type of EM data. Therefore, in order to yield accu-
rate and correct results, the method requires a rather high
image quality. The performance of this algorithm can be
judged directly by visual comparison of the extracted networks
to the raw image data and hence, a similar accuracy as
observed is expected from the subsequent orientation analysis.
Second, we will discuss a gray value gradient based orien-
tation measure based on the structure tensor calculus [25,29].
As this procedure evaluates the gray value gradients within a
small neighborhood around each pixel of the image and
extracts the desired orientation information from this calcu-
lus, it is applicable to a wide range of different images. Even if
the resolution of the image is far lower than the typical length
scale of the individual actin network constituents, this analysis
can still yield informative results. Our main conclusion is that
both methods give similar results, both for simulated and ex-
perimental data, and demonstrate clear differences between
the actin filament orientation distributions of fast and slowly
protruding parts of the lamellipodia.
MATERIALS AND METHODS
We performed experiments with motile fish keratocyte
cells as described earlier [15,28,30], but now with a special
focus on the role of growth velocity. Fish keratocytes were pre-
pared from scales of freshly killed brook trout (Salvelinus fon-
tinalis). Primary cultures of keratocytes, produced by incubat-
ing the fish scales in growth medium in plastic Petri dishes,
were treated with trypsin and the released cells re-plated onto
Formvar-coated electron microscope grids. The living cells
were imaged under phase contrast optics in a Zeiss Axioscope
inverted microscope and time lapse sequences recorded on a
Micromax CCD camera (Roper Scientific) at time intervals of
5–10 seconds. After one of the motile cells slowed down, all
cells were fixed by exchanging the growth medium for a fixa-
tion/extraction solution containing a mixture of Triton X-100
(0.75%) and glutaraldehyde (0.25%) in cytoskeleton buffer
(10 mM MES buffer, 150mM NaCl, 5mM EGTA, 5 mM glu-
cose, and 5mMMgCl2, at pH 6.1). After 1 min in this mixture,
the cells were postfixed in 2% glutaradehyde in the same
buffer containing 1lg/ml phalloidin and stored in this solu-
tion at 48C until use. For electron tomography, the grids were
negatively stained in aqueous, 4% sodium silicotungstate
(Agar Scientific) at pH 7, containing BSA-saturated colloidal
gold and tomogram series collected in an FEI Polara electron
microscope operating at 300 kV. Reprojections from the tilt
series were generated using IMOD software from the
Boulder Laboratory for 3D Electron Microscopy of Cells.
ORIGINAL ARTICLE
2 Reconstructing the Orientation Distribution of Actin Filaments
RESULTS
Live Cell Imaging and Tomography
Although we expect that changes in network growth ve-
locity trigger transitions in the filament orientation distribu-
tion in the lamellipodium, this cannot be easily manipulated
from the outside. In fact it is not sufficient to compare two
cells crawling with different absolute velocities, because the
important parameter that dictates network orientation is the
protrusion speed measured relative to the single filament poly-
merization velocity [20]. As this polymerization speed itself
depends on many intrinsic characteristics of the cell, like the
available actin monomer concentration at the leading edge
and regulatory circuits, it is an individual property of each
cell. Hence, it is not unlikely that two cells, migrating at differ-
ent speed, could still crawl in the same dynamic regime and
with similar actin network organization.
To approach this problem, we utilized two different stra-
tegies. First, we used the fact that the velocity of individual
cells crawling on a culture dish naturally varies due to random
differences in the external and internal conditions for each
cell. The analysis of correlated data from two different micros-
copy setups allowed us to exploit these stochastic variations
and extract the structural organization of the actin network
for different cell velocity and therefore protrusion speed of the
leading edge network. Different moving fish keratocyte cells
are observed in a live-microscope in real time. Once one cell
slows down significantly, all cells are fixed in their present dy-
namical state. Subsequently, the lamellipodium network of the
cells is analyzed in high resolution EM tomography. Although
it is not possible to measure directly the network velocity rela-
tive to the active single filament polymerization speed in the
final state, we know from the live-cell images which cell slowed
down just before fixation. Therefore it is likely that this cell
switched its dynamical state at this point. From here on we
can compare the network orientation distribution in this cell
to another cell which was moving unperturbed at a relatively
high velocity and compare the two for differences in the
filament orientation patterns.
Second, we used the fact that protrusion rates and there-
fore presumably also actin network organization vary consid-
erably along the boundary of a locomoting keratocyte, with a
much smaller normal component at the flanks than at the
front [31,32]. We therefore also performed analysis of EM
data from different locations at the boundary of steadily mov-
ing cells, namely both at the center of the leading edge and at
the sides.
In Figure 1, we show two snapshots from a live-micros-
copy video of motile keratocytes (see Supporting Information
movie 1). The absolute velocities of the two cells marked on the
images is not directly relevant for our considerations. However,
it is important to note that Cell 1 moved steadily and with
approximately constant velocity over the whole recording time,
while Cell 2 has slowed down to the point where its cell body
was only barely moving anymore. The important parameter for
our analysis is the relative velocity of the leading edge with
respect to filament polymerization, which we assume to be
much smaller here compared to the steadily moving cell. After
fixation and tomography reconstruction, all slices of the volu-
metric stack were preprocessed, before the actual filament net-
work analysis started. Here, the contrast of all images was
increased and bright prominent marker dots that are necessary
for the reconstruction of the tomograph were removed from
Figure 1. Two snapshots of a live-microscopy video of motile keratocyte cells at the beginning of the recorded movie (a) and just before
the fixation of the cells (b). The time code of the individual images is displayed on the respective lower right corner in units of minutes and
seconds. The two keratocyte cells that were analyzed in EM and image processed later are indicated on these images. Cell 1 moved stea-
dily over the substrate at a relatively high velocity. Compared to its initial velocity, Cell 2 slowed down significantly just before fixation.
Scale bar is 10 lm.
ORIGINAL ARTICLE
Cytometry Part A � 00A: 000�000, 2012 3
the images. Figure 2 shows a slice plot through the image stack.
It clearly demonstrates that relatively long segments of indivi-
dual filaments lie in a single image z-plane parallel to the sur-
face before they are leaving the focus of the slice (see also Sup-
porting Information movie 2). This shows that the lamellipo-
dium is essentially organized in two dimensions and validates
our approach to conduct a 2D analysis.
Filament Network Extraction
We first developed a 2D network extraction algorithm
suitable for our EM tomography slices of lamellipodium actin
networks. A similar approach has been used successfully
before to construct graphs from image data of astroglial cells
in the central nervous system [22]. Despite the fact that in
principle the network extraction routine described in the fol-
lowing could be generalized to 3D, here we restrict our
approach to two dimensions. Our available EM data consists
of volumetric stacks of 2D slices through the lamellipodium
parallel to the surface, at a position close to the leading edge
of the cell. The lamellipodium is typically 200 nm thick, but
several microns wide, and therefore effectively 2D. For the spe-
cific scope of this work, namely extracting projected filament
orientation distributions, the 2D approach is therefore suffi-
cient. To obtain other physical parameters from the extracted
actin networks, like filament connectivity, a complete 3D net-
work reconstruction would be required. Such a 3D network
analysis has been carried out before for intermediate filament
networks using a commercially available image processing
package in combination with algorithms for artifact compen-
sation [23].
The final goal of our 2D approach is to describe the com-
plex 2D actin network structures by a graph consisting of
located nodes and a connectivity matrix indicating which
pairs of nodes are directly linked by linear (filament) line seg-
ments. In this way, we are efficiently reducing the experimen-
tal data to the information which is relevant for us, the fila-
ment orientation distribution. Although nodes between seg-
ments that are found by the algorithm might often only be
artifacts but no physical crosslinks between filaments, due to
the 2D projections of the 3D network, we are interested in the
orientation of filament segments and thus this effect does not
influence our results significantly. The algorithm for network
extraction can be subdivided in the following five steps,
segmentation, skeletonization, classification of nodes, graph
creation, and network correction and simplification.
Segmentation. As a preparation for network extraction, the
actin network visible on the EM image is segmented. This is a
crucial step in the analysis as the final accuracy of this segmen-
tation will have a significant impact on the quality of the
results from later processing steps for network extraction. For
Figure 2. Plot of representative slices through a typical EM image
data cube. The total volume of the image data is 1.3 lm 3 1.2 lm3 0.1 lm.
Figure 3. Segmentation of the actin network on a representative slice of EM tomography data obtained from the lamellipodium of a motile
fish keratocyte cell. (a) Grayscale representation of one slice of the volumetric data reconstructed from EM tomography. Scale bar is
0.1lm. (b) Final black and white network image after binarization, removal of the background and smaller artifacts.
ORIGINAL ARTICLE
4 Reconstructing the Orientation Distribution of Actin Filaments
this reason it is not possible to apply this approach to low
quality image data, as then it is very hard or often even impos-
sible to reliably segment the network constituents in an auto-
mated procedure. As an alternative to an initial segmentation
step, also other approaches exist which allow to simplify a
grayscale image directly, while preserving its topology [35].
For 2D scanning electron microscope images, network charac-
teristics of intermediate filaments have been successfully ana-
lyzed along these lines before [36].
For a good segmentation, i.e., a black and white represen-
tation of the actin network on the image, we first binarized the
image using Otsu’s method [37]. This primary segmentation
however was imperfect due to artifacts outside the lamellipo-
dium and between filaments. Therefore we needed to specify
the bulk region occupied by the actin network in the image.
As the EM images were taken in close proximity to the leading
edge, not the whole image region was filled by the network.
For this purpose, we evaluated the coherency measure for the
images that have been noise reduced by median and Wiener
filters before. Coherency is able to distinguish actin network
structures from a uniform or noisy background [25]. A binar-
ized version of the coherency feature image yielded a good
estimate for the boundaries of the actin network in the 2D
slices of the volumetric image stack. As an additional measure
that is able to detect smaller holes in the actin mesh, where no
filaments are present, we used the normalized square root of
the trace of the structure tensor. This tensor had already been
calculated before to obtain the coherency. After applying a
maximum filter and subsequent binarization, we got the rele-
vant network region in the image, in which filaments were
actually present. Pixelwise multiplication of the three binar-
ized images, i.e. the one obtained from Otsu’s method, the
binarized coherency and the binarized trace of the structure
tensor yielded a preliminary result for the segmented network.
In a last step, we analyzed this image for small artifacts, i.e.,
separated clusters of pixels with a total area smaller than a
given threshold, which were then deleted. This completed the
segmentation process. A representative original image is
shown in Figure 3a, together with its segmented version in
(b). In the following, to give a detailed impression on how the
network extraction procedure performs on EM images, we will
illustrate individual analysis steps on a small region of the
large sample image. Figure 4 indicates this smaller cutout
from the sample EM slice.
Skeletonization. We begin the network extraction starting
from the black and white version of the filament network, that
is cut from the binarized full image as shown in Figure 5a.
First, a thinning or skeletonization algorithm is applied. In
this step, the width of all segmented objects is iteratively
reduced to a final thickness of one pixel, without altering the
connectivity of the binary image. The final skeletonization
result is displayed in Figure 5b.
Classification of nodes. At this point, the width of all objects
in the image is exactly one pixel and therefore it is possible to
classify endpoints of objects as nodes. Endpoints are defined
as white pixels with exactly one other white pixel in their
direct neighborhood defined by eight surrounding pixels. In
Figure 5c such endpoints are marked in red. As we disregard
single isolated white pixels, every endpoint is connected to at
least one other endpoint by a closed path of white pixels.
However, it is also possible that a single endpoint is linked to
more than one other endpoints, if the connecting white path
starting at its position splits up on the way. Hence, we define a
Figure 4. To keep the presentation of the network extraction algorithm illustrative, we will explain the analysis procedure by applying it to
a smaller sample section of the full 2D sample slice. (a) Full EM slice, with the boundary of the smaller sample region indicated in black. (b)
Magnified sample region from (a). One edge of the quadratic sample region spans 0.34 lm.
ORIGINAL ARTICLE
Cytometry Part A � 00A: 000�000, 2012 5
second class of nodes at the position of these split-ups. Such
higher order nodes or crosslinks are defined as white pixels
with at least three other white pixels in their direct neighbor-
hood. Higher order nodes are marked in green in Figure 5c.
Graph creation. Next, it is possible to construct a graph or
network by tracing connected paths sequentially starting at all
nodes which have been classified in the previous step. The
result of this procedure is essentially an adjacency matrix. This
quadratic matrix holds as many columns as nodes were found
in the previous step and all of its entries are initially set to
zero. If a connection is established between two nodes with
indices i and j, the i, j-th element of the matrix is set to one.
As links between nodes are undirected, only the upper triangle
of the symmetric matrix has to be considered. In this way, we
complement extracted nodes at certain positions in the image
with the additional information, if pairs of nodes are con-
nected. In other words, we have already extracted a primary
network from the segmented image. This rudimentary net-
work representation is shown in Figures 5d and 5e. To be able
to judge the accuracy of the extracted abstract linearized net-
work compared to the original actin filament network, the
skeletonized binary image and the sample section of the gray
value EM image are shown in the background of the plots,
respectively.
Network correction and simplification. The preliminary
network holds a rather large number of unnecessary nodes,
that does not contribute significantly to its architecture nor
the segment orientation distribution. Additionally, imperfec-
tions in the segmentation process lead to artificial disconnec-
tions in filaments. Therefore, we implemented an additional
correction and simplification routine for the primary
extracted network.
Due to the specifics of the skeletonization algorithm we
are using, multiple higher order nodes are often located in
their direct neighborhood. Merging these clusters efficiently
reduces the total number of nodes and therefore the dimen-
sion of the adjacency matrix. A new node, representative for
the whole cluster, is positioned at the centerpoint of all nodes
Figure 5. Sequential steps of the network extraction routine. (a) Sample region of the binarized 2D EM image as indicated in Figure 4. (b)
Segmented region after skeletonization. The connectivity of segmented objects is not altered in this operation. (c) Nodes are located and
classified according to the number of white pixels in their direct neighborhood. Red nodes indicate endpoints, while green nodes indicate
higher order nodes. (d) An adjacency matrix is constructed, that incorporates which nodes are directly connected by a continuous path of
white pixels. An abstract network according to straight links between connected nodes is shown in yellow. (e) The same graph as in (d) is
plotted on top of the original sample section of the gray value EM image for comparison. (f) The extracted network is simplified and cor-
rected to some extend. Unnecessary nodes, short dead ends and small loops are removed, while potential gaps in line segments are
bridged. The area of the quadratic sample region spans 0.34 lm 3 0.34 lm. [Color figure can be viewed in the online issue, which isavailable at wileyonlinelibrary.com.]
ORIGINAL ARTICLE
6 Reconstructing the Orientation Distribution of Actin Filaments
which were merged and accumulates all connections to out-
side nodes.
From Figure 5e it is also evident that the extracted net-
work incorporates many rather short dead ends. Either these
line segments do not correspond to filaments in the gray
value image at all or represent filaments that were discon-
nected due to artifacts in the original image or due to an
imperfect segmentation. Therefore, we correct for such dead
ends in the following way: First, short dead ends are detected
in the network as line segments that are shorter than a given
threshold, while connecting at least one endpoint with an ar-
bitrary node. Subsequently, the algorithm extrapolates these
segments at their endpoint and connects them to other dead
ends which are oriented similarly and located close to the
extrapolated path. In this way small gaps in filaments are
bridged. If a connection to another line segment is not likely,
the dead end is classified as an artifact and deleted from the
network.
An additional artifact of the skeletonization is the appear-
ance of very small loops in the network. Therefore, we are tra-
cing the network for such rather small loops, using a depth
first search algorithm up to a maximum segment path length.
Once a loop is detected, all involved nodes are merged to a
single node located in their center, which also accumulates all
of the connections to outside nodes.
As a last step, second order nodes, only connected to
exactly two other nodes in the network, are analyzed for their
importance. Such second order nodes might appear as a result
of one of the simplification steps. If the change in orientation
of the two connections at the position of such a node is smal-
ler than a predefined threshold, the node is removed and the
two connected nodes are linked directly.
For our example region, the final network after correction
and simplification is shown in Figure 5f. Figure 6 shows the
abstract fiber network extracted from the full sample slice of
the EM data.
Gray Value Gradient-Based Orientation Analysis
As an alternative to the network extraction algorithm, we
also implemented an orientation analysis based on the struc-
ture tensor [29]. We have utilized this approach successfully
before in the biological context to classify actin network archi-
tecture on high-throughput fluorescence microscopy data
[25]. Briefly, a strategy to extract a unit vector n defining the
local orientation in a small region of the image is to assume,
that it deviates least from the gradient direction of the image
gray values, rgð~xÞ. For this purpose, a promising procedure is
Within a finite region, this amounts to an extremum
principle,
Zþ1
�1wð~x �~x0Þ n � rgð~x0Þð Þ2 d2x0 ! max; ð2Þ
where wð~x �~x0Þ is a window function, defining the neighbor-
hood around pixel ~x ¼ ðx; yÞ. To formulate these mathemati-
cal considerations, the discrete gray values of the image were
considered as a scalar field gð~xÞ, continuous in position and
magnitude. For the analysis, we will express the necessary
mathematical operators by 2D filter masks that are suitable to
transfer our derived expressions to the quantized gray value
images. The optimization problem can be solved by rewriting
Eq. (2) as,
Figure 6. Extracted abstract fiber network as the result of the network extraction algorithm described in the main text and in Figure 5. (a)
For better judgment of the quality of the resulting network, the original gray value EM image is shown in the background. (b) Plot of the
extracted network exclusively. The scale of these images is defined as in Figure 3. [Color figure can be viewed in the online issue, which is
available at wileyonlinelibrary.com.]
ORIGINAL ARTICLE
Cytometry Part A � 00A: 000�000, 2012 7
nT Jn ! max; ð3Þwhere the structure tensor J is defined as,
Jpqð~xÞ ¼Zþ1
�1wð~x �~x0Þ @gð~x0Þ
@p
@gð~x0Þ@q
� �d2x0: ð4Þ
This is a symmetric, positive semidefinit matrix and therefore
there exist two orthogonal eigenvectors with non-negative
eigenvalues. Hence, a suitable coordinate rotation reduces J to
a diagonal matrix and the optimization problem reads,
n0x n0y� � Jx0x0 0
0 Jy 0y0
� �n0xn0y
� �! max: ð5Þ
If both eigenvalues are equal, i.e., Jx0x0 5 Jy0y0, there is no domi-
nant orientation observed in this neighborhood. Therefore,
without loss of generality, we can assume Jx0x0 [ Jy0y0 and hence
n0 ¼ 1 0½ � maximizes Eq. (5) with maximum value Jx0x0.
This means that in the coordinate system in which the struc-
ture tensor J is diagonal, i.e., the coordinate system spanned
by the two orthogonal eigenvectors of the matrix J, n and
therefore the local orientation is given parallel to the eigenvec-
tor with the largest eigenvalue.
As an example, Figure 7 shows a representative extraction
of a feature image from two different variants of a computer
generated random fiber network. These random fiber net-
works have been used before to model mechanical characteris-
tics [33] and transport properties [34] of biopolymer net-
works, but also to benchmark image processing routines for
microscopy data of actin [25]. To obtain a realization of the
stochastic network model, the positions of a constant number
of fibers are chosen uniformly randomly within a confined 2D
domain. Subsequently, the orientation angle of each fiber is
randomly selected from a specified orientation distribution.
Connections in the network are formed when two fibers hap-
pen to cross each other. For testing the extraction of fiber
orientation distributions with our approaches, the connectiv-
ity of the fibers is of minor importance for low to moderate
fiber densities; only at high fiber density, the large number of
connections makes it difficult to resolve single fibers in the
network. Reconsidering the model predictions for the struc-
tural organization of lamellipodium actin networks in [17,20],
we expect two fiber orientation patterns to be especially rele-
vant. Therefore at this point we draw fiber orientations from
two different linear combinations of Gaussian distributions
with means at either 135/235 (Figs. 7a and 7b) or 170/0/
270 degrees (Figs. 7c and 7d), each with constant standard
deviation r 5 158. In the case of 135/235 degrees networks,
the weight of both peaks is chosen equal, while for the second
170/0/270 degrees pattern, the weight of the peak at 08 is
chosen twice as large as the constant weight at 170/270
degrees. For illustration the absolute orientation of each fiber
relative to the vertical boundary of the network is color coded
in these images as indicated by the colorbar.
The lower two images in Figure 7 show results of the
structure tensor analysis applied to gray scale representations
of the network examples above using a feature image represen-
tation. The color in each pixel indicates its local orientation,
while the saturation is given by the coherency measure,
cc ¼ Jx0x0 � Jy0y 0
Jx0x0 þ Jy0y 0
� �2
¼ Jyy � Jxx� �2þ 4J 2xy
Jxx þ Jyy� �2 ; ð6Þ
and the intensity by the trace of the structure tensor,
Tr J ¼ Jxx þ Jyy ¼ Jx0x0 þ Jy0y 0 : ð7Þ
This is a variant of the HSV-colorspace with a constraint hue
domain. Colored pixels in the image indicate regions of clear
orientation, while black pixels mark the uniform background
without any dominant orientation. With regard to visual
inspection the feature images appear to capture the original
fiber orientation of their individual sample images reasonably
well. In the following section, we will quantify this impression
by calculating fiber orientation distributions from such
random fiber networks and compare them to the actual
Figure 7. Orientation analysis of two realizations of random fiber
networks using the structure tensor calculus. (a) and (c): Two
examples of random fiber networks according to a 135/235 (a)and a 170/0/270 degrees (c) distribution. The orientation of theindividual filaments is color labeled. (b) and (d): Feature images
of the orientation, that were extracted using the structure tensor
analysis of black and white versions of (a) and (c). The colorbars
shown apply to all four images. To obtain such illustrative exam-
ple images, the size of the window function (i.e., Gaussian filter)
was reduced compared to the later analysis to obtain relatively
sharp feature representations of the individual fibers. [Color fig-
ure can be viewed in the online issue, which is available at
wileyonlinelibrary.com.]
ORIGINAL ARTICLE
8 Reconstructing the Orientation Distribution of Actin Filaments
orientation probability distributions from which the ensemble
of fibers has been drawn.
Analysis of Simulated Random Fiber Networks
To benchmark the accuracy of our two different image
processing routines for extracting filament orientation pat-
terns, we processed the two realizations of the computer gen-
erated random fiber model introduced earlier in ‘‘Gray Value
Gradient-Based Orientation Analysis’’ Section. The line of ref-
erence for color labeling the fiber orientations in Figures 8a
and 8d is assumed vertically, parallel to the lateral boundaries
of the network like before. As can be roughly estimated by eye
from this color labeling, the network in (a) was randomly cre-
ated to simulate a 135/235 degrees pattern, while a comple-
mentary 170/0/270 degrees distribution is visible in (d). In
(b) and (e) the extracted filament networks for the two exam-
ples are shown. The network extraction procedure was applied
to a binarized version of the random fiber networks in (a) and
(d) and subsequently extracted filament segments were color
coded according to their individual absolute orientation.
Extracted orientation distributions for the two simulated net-
works from our two independent approaches are shown in (c)
and (f). To evaluate this distribution from an extracted net-
work, the orientation of each line was considered as a small
Gaussian with standard deviation r 5 38 weighted by the
corresponding segment length. In case of the structure tensor
the evaluated orientation at each pixel was considered and
weighted by the coherency measure cc at this position. The full
distribution was subsequently convolved with a Gaussian filter
of size 128 and r 5 48 to average out fluctuations. All applied
methods are able to recover the original fiber orientation dis-
tribution of the random networks to a similar, reasonably high
extend. The major difficulty arises in the large angle domain
of the 170/0/270 degrees pattern. This is due to the limited
separation of the 170/270 degrees peaks. As the orientation
of filaments is only defined in an angle domain modulo 1808,the orientation difference between these peaks is only 408. Itturns out, that the resolution of the applied analysis is not suf-
ficiently high to clearly resolve those two separated Gaussians.
Analysis of Lamellipodia Networks
Finally, the two independent automated orientation anal-
ysis routines, i.e. network extraction and gradient based struc-
ture tensor calculus, were applied to stacks of EM images
obtained experimentally. In case of the structure tensor rou-
tine, the preprocessed images were filtered with a median filter
to reduce noise before analysis. Results from network extrac-
tion with color labeled orientation relative to a line perpendic-
ular to the leading edge are shown in Figures 9b and 9e. The
leading edge orientation is indicated as a separate black dashed
Figure 8. Orientation analysis of random fiber networks using network extraction and structure tensor calculus. (a) and (d): Two realiza-
tions of the random fiber networks. The model orientation distributions are either peaked at135/235 (a) or 170/0/270 degrees (d), as indi-cated by the black curve in (c) and (f), respectively. (b) and (e): Extracted filament networks for the two sample networks above. (c) and (f):
Resulting filament orientation distributions extracted from the networks according to our two independent approaches, using the structure
tensor analysis (red line) and the network extraction routine (dashed blue line). The true distribution that was used during random fiber
creation is shown as a benchmark (black line). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
ORIGINAL ARTICLE
Cytometry Part A � 00A: 000�000, 2012 9
line. This leading edge orientation was fitted manually to the
stack and held as a constant reference orientation for the com-
plete stack.
The final results for the orientation distribution of the
filaments in the lamellipodium networks are shown in Figures
9c and 9f for Cell 1 and Cell 2 (compare Fig. 1). To obtain
these distributions, we averaged over twenty adjacent slices in
the middle of each stack, where the lamellipodium network
appeared to be best visible without artifacts from the upper or
lower boundaries. The tomograph voxels are cubic and so the
spacing in z-direction is equal to the lateral dimensions, 0.746
nm. Therefore, in total we are averaging over around 14.92
nm in z. Considering typical actin filament diameters around
7 nm, this means that we take into account filaments from at
least two independent layers of actin and so the results are not
only locally representative but rather for a reasonable fraction
of the bulk network. This analysis shows, that the actin net-
work protruding in the steadily moving Cell 1 features a pro-
minent orientation pattern around 135/235 degrees. In
marked contrast, Cell 2, which has slown down before fixa-
tion, yields a qualitatively different distribution. First, this
orientation pattern spreads to broader angles more parallel to
the front as already indicated in [28] and second, a single peak
is observed within an angle domain near perpendicular to the
leading edge. Although peaks at 170/270 degrees are not
detected, as would be expected from the theoretical predic-
tions [20], this might be attributed to the close proximity of
these populations in orientation space modulo 1808 as it wasobserved for the analysis of artificially created random fiber
networks before in Figure 8 (‘‘Analysis of Simulated Random
Fiber Networks’’ section). In general, the 135/235 degrees
pattern is expected to always give a clearer signature than the
170/270 degrees pattern.
The results from the two independent image analysis pro-
cedures shown in Figure 9 demonstrate a clear difference in
the orientation patterns of the fast and slow moving cells.
However, there are also clear differences between the results
from the different algorithms, especially for the slower cell at
large filament orientations. The strong asymmetry of the
orientation distribution extracted with the structure tensor
might have different reasons, including a stochastic fluctua-
tion in network growth (compare the simulated realizations in
Figure 8, which also lead to asymmetries), the corrugation in
the nearby cell edge, or some artifact resulting from the algo-
rithm. In general, the two procedures developed here are very
different in nature. Although the network extraction routine
Figure 9. Representative slices of the EM tomograph and subsequent analysis results. The upper images correspond to Cell 1 (fast) and
the lower images to Cell 2 (slow) as indicated in Figure 1. (a) and (d): Representative slices of the original data stack after preprocessing.
Scale bar is 0.1 lm. (b) and (e): Extracted 2D filament segments from these images plotted in color coded orientation. (c) and (f): Orienta-
tion distributions of the two independent analysis methods averaged over twenty slices of the complete stack (i.e., 20 3 0.746 nm 514.92 nm). Results from the structure tensor calculus are red and from filament network extraction are dashed blue. [Color figure can be
viewed in the online issue, which is available at wileyonlinelibrary.com.]
ORIGINAL ARTICLE
10 Reconstructing the Orientation Distribution of Actin Filaments
exclusively accounts for the filament orientation of segments
that have been explicitly identified due to their large contrast,
the structure tensor averages orientations over all grayvalue
gradients within the network region. Future work is required
to determine which of the two algorithms gives a more reliable
presentation of the actin filament orientations, but both agree
in demonstrating a clear difference between fast and slow cells.
The same results were obtained for EM data sets taken
from different locations along the cell boundary of steadily
moving cells (additional data given in Supporting Informa-
tion). Three additional data sets were analyzed from the front
of the lamellipodium. These networks feature the same criss-
cross filament orientation pattern as the steadily moving Cell
1 in Figure 9. Then two additional data sets were analyzed
from the lateral flanks of the lamellipodium, where protrusion
in the locally orthogonal direction is diminished. Here, quali-
tatively different filament orientation distributions emerge,
which feature a single dominant peak approximately orthogo-
nal to the local orientation of the cell membrane. Therefore
slowly protruding parts of the lamellipodium of a fast moving
cell are organized in a similar manner than the lamellipodium
at the leading edge of the slow moving cell in Figure 9.
DISCUSSION
The actin filament orientation distribution is a key quantity
to understand how the lamellipodium performs its function.
Here we have described a quantitative approach to extract this
orientation distribution for the lamellipodium of locomoting ker-
atocytes and found that two distinctly different patterns emerge
for fast versus slowly protruding networks. In particular, we find
that the leading edge of slowly moving cells and the flanks of fast
moving cells show the same broad orientation distribution, in
contrast to the leading edge of fast moving cells, which show the
well known criss-cross pattern of fast growing networks.
We have developed two independent data analysis meth-
ods for the orientation analysis of the lamellipodium EM data.
Both procedures work mostly automated with only minor or
no manual adjustments necessary. On the microscopic level,
we have implemented a network extraction algorithm which
generates abstract graph representations of the underlying
actin networks. The quality of the resulting artificial networks
can be tested directly by visual comparison to the original gray
value EM images. On a more macroscopic level, we have used
an established gradient based structure tensor analysis. This
method does not incorporate any a priori knowledge that
could bias the resulting orientation distributions.
We tested our analysis with artificially created random
fiber networks for the two different orientation patterns that
have been theoretically predicted before. The two approaches
were able to reliably distinguish the two patterns and extract a
reasonable estimate for the underlying original distribution.
Applying our analysis to experimental data, we measured and
analyzed actin networks in the lamellipodia of two different
fish keratocyte cells. One was moving steadily while the other
slowed down just before fixation. Both independent analysis
methods yielded similar results on each of these two sample
cells. The resulting orientation distributions were averaged
over 20 subsequent slices of the volumetric EM data. For the
fast cell, a dominant distribution with peaks around 135/235
degrees was measured, which could be attributed to a cell
moving in a medium growth phase dynamic regime as pre-
dicted theoretically in [20]. The second perturbed cell featured
a broad distribution with a dominant peak at an orientation
almost orthogonal to the leading edge. This hints at a cell
moving in a different regime, termed slow growth phase in the
theoretical description. Secondary peaks at 170/270 degrees
which would be expected from the model were not observed.
However, this could be attributed to the limited resolution of
the two methods as confirmed by the analysis of artificial ran-
dom fiber networks. Our conclusions were verified by an addi-
tional analysis using EM data sets from different locations
along the boundary of steadily moving cells. This showed that
the slowly protruding regions at the flanks of fast moving cells
have a similar actin orientation distribution like the front
regions in slowly moving cells. Future experiments have to
show if the obtained results can be confirmed for larger cell
populations.
Our representative orientation analysis results fit very
well to previously published experimental data, where changes
in the filament orientation of mouse melanoma cells have
been correlated to the speed of lamellipodium protrusion
[28]. In this study, it was also observed that the orientation
distribution broadens for slower velocities of the leading edge.
The dominant 135/235 degrees pattern, that has been
observed in the steadily moving cell, has also been reported
previously in Refs. 17 and 26. The methods developed here
now set the ground to evaluate the actin filament orientation
distribution in migrating cells for a larger number of data sets
and for systematically varied conditions. In the long run, this
might allow us to clearly discriminate between different model
predictions.
ACKNOWLEDGMENTS
The authors thank Stefan Kostler and Anna Kreshuk for
their fruitful discussions. USS is a member of the Cell-
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12 Reconstructing the Orientation Distribution of Actin Filaments